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Estimating C4 photosynthesis parameters by fitting intensive A/Ci curves


Measurements of photosynthetic assimilation rate as a function of intercellular CO2 (A/Ci curves) are widely used to estimate photosynthetic parameters for C3 species, yet few parameters have been reported for C4 plants, because of a lack of estimation methods. Here, we extend the framework of widely used estimation methods for C3 plants to build estimation tools by exclusively fitting intensive A/Ci curves (6–8 more sampling points) for C4 using three versions of photosynthesis models with different assumptions about carbonic anhydrase processes and ATP distribution. We use simulation analysis, out of sample tests, existing in vitro measurements and chlorophyll-fluorescence measurements to validate the new estimation methods. Of the five/six photosynthetic parameters obtained, sensitivity analyses show that maximal-Rubisco-carboxylation-rate, electron-transport-rate, maximal-PEP-carboxylation-rate, and carbonic-anhydrase were robust to variation in the input parameters, while day respiration and mesophyll conductance varied. Our method provides a way to estimate carbonic anhydrase activity, a new parameter, from A/Ci curves, yet also shows that models that do not explicitly consider carbonic anhydrase yield approximate results. The two photosynthesis models, differing in whether ATP could freely transport between RuBP and PEP regeneration processes yielded consistent results under high light, but they may diverge under low light intensities. Modeling results show selection for Rubisco of low specificity and high catalytic rate, low leakage of bundle sheath, and high PEPC affinity, which may further increase C4 efficiency.

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a :

Light absorptance of leaf

A c :

Rubisco carboxylation assimilation rate


RuBP carboxylation and PEPc carboxylation limitation assimilation

RrPc :

RuBP regeneration and PEP carboxylation limitation assimilation

A g :

Gross CO2 assimilation rate per unit leaf area

A j :

RuBP regeneration assimilation rate

A n :

Net CO2 assimilation rate per unit leaf area

RcPr :

RuBP carboxylation and PEPc regeneration limitation assimilation

RrPr :

RuBP regeneration and PEPc regeneration limitation assimilation


The fraction of O2 evolution occurring in the bundle sheath

c :

Scaling constant for temperature dependence for parameters

CaL :

Lower boundary CO2 under which assimilation is limited by RuBP carboxylation and PEPc carboxylation

CaH :

Higher boundary CO2 above which assimilation is limited by RuBP regeneration and PEPc regeneration

C bs :

Bundle sheath CO2 concentration

C i :

Intercellular CO2 concentration

C m :

Mesophyll CO2 concentration

ΔH a :

Energy of activation for temperature dependence for parameters

ΔH d :

Energy of deactivation for temperature dependence for parameters

ΔS :

Entropy for temperature dependence for parameters

\(\phi_{{_{\text{PSII}} }}\) :

Quantum yield


The specificity of Rubisco at 25 °C

g bs :

Bundle sheath conductance for CO2

g bso :

Bundle sheath conductance for O2

g m :

Mesophyll conductance for CO2

I :

Light intensity


Maximum rate of electron transport at 25 °C at a specific light intensity

J max(25):

Maximum rate of electron transport at 25 °C

K c(25):

Michaelis–Menten constant of Rubisco activity for CO2 at 25 °C

K o(25):

Michaelis–Menten constants of Rubisco activity for O2

K p(25):

Michaelis–Menten constants of PEP carboxylation for CO2

O bs :

O2 concentration in the bundle sheath cells

Q10 for K p :

Temperature sensitivity parameter for Kp

R :

The molar gas constant

R d :

Daytime respiration

R dbs :

Daytime respiration in bundle sheath cells

R dm :

Daytime respiration in mesophyll cells


Ribulose-1,5-bisphosphate carboxylase/oxygenase



T k :

Leaf absolute temperature

V c :

Velocity of Rubisco carboxylation

V cmax(25):

Maximal velocity of Rubisco carboxylation at 25 °C

V p :

PEP carboxylation

V pc :

PEPc reaction rate

V pmax(25):

Maximal PEP carboxylation rate at 25 °C

V pr :

PEP regeneration rate

x :

The maximal ratio of total electron transport could be used for PEP carboxylation


  1. Baker NR, Harbinson J, Kramer DM (2007) Determining the limitations and regulation of photosynthetic energy transduction in leaves. Plant Cell Environ 30:1107–1125

  2. Barbour MM, Evans JR, Simonin KA, Von Caemmerer S (2016) Online CO2 and H2O oxygen isotope fractionation allows estimation of mesophyll conductance in C4 plants, and reveals that mesophyll conductance decreases as leaves age in both C4 and C3 plants. New Phytol 210:875–889

  3. Bellasio C, Burgess SJ, Griffiths H, Hibberd JM (2014) A high throughput gas exchange screen for determining rates of photorespiration or regulation of C4 activity. J Exp Bot 65:3769–3779

  4. Bellasio C, Beerling DJ, Griffiths H (2015) Deriving C4 photosynthetic parameters from combined gas exchange and chlorophyll fluorescence using an Excel tool: theory and practice. Plant Cell Environ. https://doi.org/10.1111/pce.12626

  5. Boyd RA, Gandin A, Cousins AB (2015) Temperature responses of C4 photosynthesis: biochemical analysis of Rubisco, phosphoenolpyruvate carboxylase, and carbonic anhydrase in Setaria viridis. Plant Physiol 169:1850–1861

  6. Brown RH, Byrd GT (1993) Estimation of bundle sheath cell conductance in C4 species and O2 insensitivity of photosynthesis. Plant Physiol 103:1183–1188

  7. Chi Y, Xu M, Shen R, Yang Q, Huang B, Wan S (2013) Acclimation of foliar respiration and photosynthesis in response to experimental warming in a temperate steppe in northern China. PLoS ONE 8:e56482

  8. Cousins AB, Baroli I, Badger MR, Ivakov A, Lea PJ, Leegood RC, von Caemmerer S (2007) The role of phosphoenolpyruvate carboxylase during C4 photosynthetic isotope exchange and stomatal conductance. Plant Physiol 145:1006–1017

  9. Cousins AB, Ghannoum O, von Caemmerer S, Badger MR (2010) Simultaneous determination of Rubisco carboxylase and oxygenase kinetic parameters in Triticum aestivum and Zea mays using membrane inlet mass spectrometry. Plant Cell Environ 33:444–452

  10. Dubois JB, Fiscus EL, Booker FL, Flowers MD, Reid CD (2007) Optimizing the statistical estimation of the parameters of the Farquhar-von Caemmerer-Berry model of photosynthesis. New Phytol 176:402–414

  11. Ethier GJ, Livingston NJ, Harrison DL, Black TA, Moran JA (2006) Low stomatal and internal conductance to CO2 versus Rubisco deactivation as determinants of the photosynthetic decline of ageing evergreen leaves. Plant Cell Environ 29:2168–2184

  12. Farquhar GD, von Caemmerer S, Berry JA (1980) A biochemical model of photosynthetic carbon dioxide assimilation in leaves of 3-carbon pathway species. Planta 149:78–90

  13. Galmés J, Hermida-Carrera C, Laanisto L, Niinemets U (2016) A compendium of temperature responses of Rubisco kinetic traits: variability among and within photosynthetic groups and impacts on photosynthesis modeling. J Exp Bot 67:5067–5091

  14. Genty B, Briantais J, Baker N (1989) The relationship between the quantum yield of photosynthetic electron transport and quenching of chlorophyll fluorescence. Biochim Biophys Acta 990:87–92

  15. Gu L, Pallardy SG, Tu K, Law BE, Wullschleger SD (2010) Reliable estimation of biochemical parameters from C3 leaf photosynthesis-intercellular carbon dioxide response curves. Plant Cell Environ 33:1852–1874

  16. Hatch MD (1987) C4 photosynthesis: a unique blend of modified biochemistry, anatomy and ultrastructure. Biochim Biophys Acta 895:81–106

  17. Hatch MD, Burnell JN (1990) Carbonic anhydrase activity in leaves and its role in the first step of C4 photosynthesis. Plant Physiol 93:825–828

  18. Heckmann D, Schulze S, Denton A, Gowik U, Westhoff P, Weber AP, Lercher MJ (2013) Predicting C4 photosynthesis evolution: modular, individually adaptive steps on a Mount Fuji fitness landscape. Cell 153:1579–1588

  19. Jenkins CLD, Furbank RT, Hatch MD (1989) Mechanism of C4 photosynthesis. A model describing the inorganic carbon pool in bundle-sheath cells. Plant Physiol 91:1372–1381

  20. Kromdijk J, Ubierna N, Cousins AB, Griffiths H (2014) Bundle-sheath leakiness in C4 photosynthesis: a careful balancing act between CO2 concentration and assimilation. J Exp Bot 65:3443–3457

  21. Miao ZW, Xu M, Lathrop RG, Wang YF (2009) Comparison of the A-Cc curve fitting methods in determining maximum ribulose 1.5-bisphosphate carboxylase/oxygenase carboxylation rate, potential light saturated electron transport rate and leaf dark respiration. Plant Cell Environ 32:1191–1204

  22. Osborne CP, Beerling DJ (2006) Nature’s green revolution: the remarkable evolutionary rise of C4 plants. Philos Trans R Soc B 361:173–194

  23. Osborne CP, Sack L (2012) Evolution of C4 plants: a new hypothesis for an interaction of CO2 and water relations mediated by plant hydraulics. Philos Trans R Soc B 367:583–600

  24. Pedomo JA, Cavanagh AP, Kubien DS, Galmes J (2015) Temperature dependence of in vitro Rubisco kinetics in species of Flaveria with different photosynthetic mechanisms. Photosynth Res 124:67–75

  25. Pinto H, Sharwood RE, Tissue DT, Ghannoum O (2014) Photosynthesis of C3, C3–C4, and C4 grasses at glacial CO2. J Exp Bot 65:3669–3681

  26. Sage RF (2002) Variation in the kcat of Rubisco in C3 and C4 plants and some implications for photosynthetic performance at high and low temperature. J Exp Bot 53:609–620

  27. Savir Y, Noor E, Milo R, Tlusty T (2010) Cross-species analysis traces adaptation of Rubisco toward optimality in a low-dimensional landscape. Proc Natl Acad Sci USA 107:3475–3480

  28. Sharkey TD, Bernacchi CJ, Farquhar GD, Singsaas EL (2007) Fitting photosynthetic carbon dioxide response curves for C3 leaves. Plant Cell Environ 30:1035–1040

  29. Studer RA, Christin PA, Williams MA, Orengo CA (2014) Stability-activity tradeoffs constrain the adaptive evolution of Rubisco. Proc Natl Acad Sci USA 111:2223–2228

  30. Ubierna N, Sun W, Kramer DM, Cousins AB (2013) The efficiency of C4 photosynthesis under low light conditions in Zea mays. Miscanthus X giganteus and Flaveria bidentis. Plant Cell Environ 36:365–381

  31. Ubierna N, Gandin A, Boyd RA, Cousins AB (2017) Temperature response of mesophyll conductance in three C4 species calculated with two methods: 18O discrimination and in vitro Vpmax. New Phytol 214:66–80

  32. von Caemmerer S (2000) Biochemical models of photosynthesis. In: Techniques in plant sciences. CSIRO Publishing, Colingwood, p 196

  33. von Caemmerer S, Quinn V, Hancock NC, Price GD, Furbank RT, Ludwig M (2004) Carbonic anhydrase and C4 photosynthesis: a transgenic analysis. Plant Cell Environ 27:697–703

  34. Xu LK, Baldocchi DD (2003) Seasonal trends in photosynthetic parameters and stomatal conductance of blue oak (Quercus douglasii) under prolonged summer drought and high temperature. Tree Physiol 23:865–877

  35. Yin X, Struik PC, Romero P, Harbinson J, Evers JB, Van Der Putten PEL, Vos JAN (2009) Using combined measurements of gas exchange and chlorophyll fluorescence to estimate parameters of a biochemical C3 photosynthesis model: a critical appraisal and a new integrated approach applied to leaves in a wheat (Triticum aestivum) canopy. Plant Cell Environ 32:448–464

  36. Yin X, Sun Z, Struik PC, Gu J (2011a) Evaluating a new method to estimate the rate of leaf respiration in the light by analysis of combined gas exchange and chlorophyll fluorescence measurements. J Exp Bot 62:3489–3499

  37. Yin XY, Sun ZP, Struik PC, Van der Putten PEL, Van Ieperen W, Harbinson J (2011b) Using a biochemical C4 photosynthesis model and combined gas exchange and chlorophyll fluorescence measurements to estimate bundle-sheath conductance of maize leaves differing in age and nitrogen content. Plant Cell Environ 34:2183–2199

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We are grateful for support from the University of Pennsylvania. We thank Dr. Jesse Nippert, Kansas State University, for providing the fluorometer chamber.


We sincerely thank the constructive comments from two anonymous reviewers. The experiments are supported by Department Research Fund (to H.Z.) from Department of Biology, University of Pennsylvania.

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Correspondence to Haoran Zhou.

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Electronic supplementary material

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Supplementary material 1 (XLSX 56 KB) Supplementary Material I. Deriving C4 photosynthesis parameters by fitting A/Ci curves for model without carbonic anhydrase reaction using Sharkey’s fitting procedure.

Supplementary material 2 (XLSX 67 KB) Supplementary Material II. Deriving C4 photosynthesis parameters by fitting A/Ci curves for model without carbonic anhydrase reaction using Yin’s fitting procedure.

Supplementary material 3 (XLSX 59 KB) Supplementary Material III. Deriving C4 photosynthesis parameters by fitting A/Ci curves for model with carbonic anhydrase reaction using Sharkey’s fitting procedure.

Supplementary material 4 (DOCX 26768 KB) Supplementary Material IV. Estimation results for two estimation methods of with/without carbonic anhydrase reaction for nine species (using Supplementary Material I and III).

Supplementary material 5 (XLSX 34 KB) Supplementary Material V. Instruction for use and set the solver macro.

Supplementary material 6 (XLSX 49 KB) Supplementary Material VI. Resources and data for Temperature dependence of input and output parameters.

Supplementary material 7 (DOCX 156 KB) Supplementary Methods, Tables and Figures.

Supplementary material 8 (R 10 KB) C4Estimation 0.1.tar.gz is the R package which contains estimation methods for model with and without carbonic anhydrase reaction using Sharkey and Yin’s fitting procedures (same with Supplementary Material I, II and III) and three additional methods in which researchers can provide new temperature response parameters further.

Supplementary material 9 (RD 2 KB)

Supplementary material 10 (R 14 KB)

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Supplementary material 12 (3D 9 KB)

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Supplementary material 14 (R 12 KB)

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Supplementary material 16 (RWL 11 KB)

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Zhou, H., Akçay, E. & Helliker, B.R. Estimating C4 photosynthesis parameters by fitting intensive A/Ci curves. Photosynth Res 141, 181–194 (2019). https://doi.org/10.1007/s11120-019-00619-8

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  • A/C i curves
  • C4
  • Estimation method
  • Non-linear curve fitting
  • Photosynthesis parameters
  • V cmax
  • Electron transport
  • PEP carboxylation rate
  • Carbonic anhydrase